Make custom post type column sortable

My site has two custom types: Auction and Lot. Lots are connected to an Auction via the post_parent field. I added an Auction column to the Lots list in the admin area.

function theme_lot_custom_columns($columns){
    $new = array();
    foreach($columns as $key => $value) {
        if ($key=='date') {
            // Put the Auction column before the Date column
            $new['auction'] = __('Auction');
        }
        $new[$key] = $value;
    }
return $new;
}
add_filter('manage_lot_posts_columns', 'theme_lot_custom_columns');

function theme_lot_custom_column($column, $post_id) {
    if ($column === 'auction'){
        $post = get_post($post_id);
        if ($post->post_parent) {
            $post_parent = get_post($post->post_parent);
            echo '<a href="' . get_edit_post_link($post_parent->ID). '">' . $post_parent->post_title . '</a>';
        }
    }
}
add_action('manage_lot_posts_custom_column', 'theme_lot_custom_column', 5, 2);

I’d like to make this Auction column sortable alphabetically by the auction title. I thought that adding the following code was enough, but no.

function theme_lot_sortable_columns($columns) {
    $columns['auction'] = 'auction';
    return $columns;
}   
add_filter('manage_edit-lot_sortable_columns', 'theme_lot_sortable_columns');

Any help would be appreciated.

Thanks!

add_action( 'pre_get_posts', 'as25_column_orderby' );
function as25_column_orderby( $orderby ) {
 if( ! is_admin() || 'lot' != $query->get( 'post_type' ) )
    return;

 $orderby = $query->get( 'orderby' );

  if( 'auction' == $orderby ) {
   $query->set( 'orderby', 'parent' );
  }
}

function sort_column_by_auction_1357( $vars ){
if ( isset( $vars["orderby"] ) && "auction" == $vars["orderby"] ) {
    $vars = array_merge( $vars, array(
        "orderby" => "auction"
    ) );
}
return $vars;
}
add_filter( "request", "sort_column_by_auction_1357" );